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Submitted on 14 Dec 2014
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Dimensional contraction in Wasserstein distance for
diffusion semigroups on a Riemannian manifold
Ivan Gentil
To cite this version:
Dimensional contraction in Wasserstein distance for diffusion
semigroups on a Riemannian manifold
Ivan Gentil
∗December 14, 2014
Abstract
We prove a refined contraction inequality for diffusion semigroups with respect to the Wasser-stein distance on a compact Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional contraction established in [BGG13] for the Euclidean heat equation. It is proved by using a dimensional coercive estimate for the Hodge-de Rham semigroup on 1-forms.
Key words: Diffusion equations, Wasserstein distance, Hodge-de Rham operator, Curvature-dimension bounds.
Mathematics Subject Classification (2000): 58J65, 58J35, 53B21
1
Introduction
The von Renesse-Sturm Theorem (c.f. [vRS05]) insures that the contraction of the heat equation on a Riemannian manifold with respect to the Wasserstein distance is equivalent to a uniform lower bound of the Ricci curvature. This result is one of the first equivalence theorems relating the Wasserstein distance and the Ricci curvature. Actually, there are many extensions of this result including the case of a heat equation on a metric measure space and we would like to take account of the dimension into such contraction.
Let us explain the contraction inequality and its extensions with more details. For simplicity, we focus on the heat equation on a Riemannian manifold but all of these results have been proved for a general diffusion semigroup on a Riemannian manifold (or more general spaces). Let ∆ be the Laplace-Beltrami operator on a smooth Riemannian manifold (M, g) and let Ptf be the
solution of the heat equation ∂tu = ∆u with f as the initial condition.
The von Renesse-Sturm Theorem states that: Let R ∈ R, the following assertions are equiv-alent (the Wasserstein distance is denoted W2),
(i) for any f, g probability densities with respect to the Riemannian measure dx W22(Ptf dx, Ptgdx) ≤ e−2RtW22(f dx, gdx), ∀t > 0,
∗Institut Camille Jordan, Umr Cnrs 5208, Universit´e Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918,
(ii) Riccig > R (uniformly in M ) where Riccig is the Ricci tensor of (M, g). The inequality
has to be understood in the sense of inequality between symmetric tensors.
There are many proofs and extensions of this result, one can see for instance [Wan04, OW05, AGS08, Kuw10, Wan11, Sav14, GKO13, BGL14, BGL15].
Recently, many extensions have been given taking account of the dimension of the manifold. For instance in [Kuw13, BGL15] the authors prove that if M is a n-dimensional Riemannian manifold with a non-negative Ricci curvature, then for any f, g probability densities with respect to dx, and any s, t > 0,
W22(Psf dx, Ptgdx) ≤ W22(f dx, gdx) + 2n(
√
s −√t)2, ∀s, t > 0, (1) Non-negative curvature condition has been removed in [Kuw13, EKS13]. If M is a n-dimensional Riemannian manifold, the main extensions are the following:
• In [Kuw13], K. Kuwada proves that the Ricci curvature is bounded from below by R ∈ R if and only if for every s, t > 0,
W22(Ptf dx, Psgdx) ≤ A(s, t, R)W22(f dx, gdx) + B(s, t, n, R), ∀s, t > 0, (2)
for probability densities f, g with respect to dx, for appropriate functions A, B > 0. • In [EKS13], M. Erbar, K. Kuwada and K.-T. Sturm prove that the Ricci curvature of the
n-dimensional manifold M is bounded from below by a constant R ∈ R if and only if sR n 1 2W2(Ptf dx, Psgdx) 2 ≤ e−R(t+s)sR n 1 2W2(f dx, gdx) 2 + n R(1 − e −R(s+t))( √ t −√s)2 2(t + s) , (3) for any s, t > 0 and any probability densities f and g. Here sr(x) = sin(√rx)/√r if r > 0,
sr(x) = sinh(√−rx)/√−r if r < 0 and s0(x) = x, hence recovering (1) when R = 0.
• In [BGG13] we prove that the classical heat equation in the Euclidean space Rn satisfies, for any f , g probability densities with respect to the Lebesgue measure λ,
W22(Ptf λ, Ptgλ) ≤ W22(f λ, gλ) −
2 n
Z t
0
Entλ(Puf ) − Entλ(Pug)2du, ∀t > 0, (4)
where Ent is the Entropy (it will be defined later).
• In the same way, again in [BGG13], we prove a more general result for a n-dimensional Riemannian manifold with a Ricci curvature bounded form below by R and for the Markov transportation distance T2 (a new distance on measures). We obtain for any f and g,
T22(Ptf dx, Ptgdx) ≤ e−2RtT22(f dx, gdx) −
2 n
Z t
0
e−2R(t−u)(Entµ(Pug) − Entµ(Puf ))2du,
for any t > 0.
The main result of this paper can be stated as follows, let dµ = e−Ψdx be a probability measure with Ψ a smooth function on M (for the sequel the manifold will be compact). We denote by (Pt)t>0 the Markov semigroup associated to the generator L = ∆ − ∇Ψ · ∇. Then
under the curvature-dimension condition
Riccig+ Hess(Ψ) > R +
1
m − n∇Ψ ⊗ ∇Ψ, (5)
for some R ∈ R and m > n (when m = n then Ψ = 0), then for any probability densities f, g and any t > 0, W22(Ptf µ, Ptgµ) ≤ e−2RtW22(f µ, gµ) − 2 m Z t 0
e−2R(t−u)Entµ(Pug) − Entµ(Puf )
2 du. The main advantage of such inequality with respect to (2) and (3) is to obtain a contraction inequality with the same time t instead two different times s and t. Moreover, the additional term is given with a minus sign which shows the improvement given by the dimension.
The method to get a dimensional contraction is radically different that one one used in [EKS13]. Here the strategy to prove such inequality is the Benamou-Brenier dynamical formulation (the Eulerian formulation) of the Wasserstein distance associated to a sharp dimensional estimate on the Hodge-de Rham semigroup. The strategy is closed to the one used in [OW05]. In [EKS13] the authors use in force the definition of the heat equation as a gradient flow of the entropy with respect to the Wasserstein distance.
The paper is organized as follow. First in Section 2, we recall the Riemannian setting and the Wasserstein distance through the Benamou-Brenier dynamical formulation. We need to introduce the Hodge-de Rham operator on forms and its associated semigroup. In Section 3, we improve the Bochner-Lichnerowicz-Weitzenb¨ock identity for 1-forms to get a coercive inequality for the Hodge-de Rham semigroup. Finally in Section 4 the main theorem is proved.
For simplicity reasons, the result will be stated and proved in the context of a compact Riemannian manifold but its generalization in a metric space, including the equivalence with respect to the condition (5), is actually a project with F. Bolley, A. Guillin and K. Kuwada. Moreover, the main theorem is written for a reversible semigroup but the proof can be adapted to the non-reversible case. In that case the solution of the heat equation is not a gradient flow of the entropy with respect to the Wasserstein distance and the method proposed in [EKS13] can not be applied.
2
Framework and main result
2.1
Geometrics tools
Conventions and notations. Let (M, g) be a n-dimensional, connected, compact and differ-entiable Riemannian manifold without boundary. We assume for simplicity that the manifold is C∞. For each x ∈ M we denote by TxM the tangent space to M and by T M the whole
tangent bundle of M . Moreover gx is a symmetric definite positive quadratic form on TxM , in
a local basis of TxM (ei)1≤i≤n, gx = (gij). (For simplicity, the x dependance of the metric g
measure. In the sequel we will use the Einstein convention of summation over repeated indices: for instance xiyi=Pni=1xiyi, xigijyj =Pni,j=1xigijyj.
The Riemannian scalar product between two tensors X and Y is denoted X ·Y , its associated norm is noted |X| (depending on g). For instance, locally in a basis (ei)1≤i≤n for two smooth
functions f, h : M 7→ R, ∇f · ∇h = ∂if gij∂jh where (gij) = (gij)−1. As usual the covariant
derivative in the direction ei of a tensor X (vector field or form) is noted ∇iX. The geometric
musicology will be used in force, for instance ∇if = gij∇jf = gij∂jf or ∇iX = gij∇jX. If ω is
a 1-form then ω∗ is its dual representation as a vector field with components ωi = gijω j. The
Ricci tensor of (M, g) is noted Riccig.
The Laplace-Beltrami operator ∆ is acting on smooth functions f , ∆f = ∇ · ∇f,
where ∇· is the divergence operator acting on vector fields, ∇ · X = ∇iXi for every vector field
X. (We use the analyst’s convention with respect to the sign.) A smooth function (or form) is a C∞-function (or form). The divergence operator ∇· satisfies for any smooth vector fields X and any smooth functions f ,
Z
M∇ · Xf dx = −
Z
M∇f · Xdx.
The Laplace-Betrami operator can also be written as ∆f = δdf where δ is the divergence operator on 1-form
δω = ∇iωi = gij∇iωj = gij(∂iωj− Γpijωp),
where Γpij are Christoffel symbols. Operators δ and ∇· are related by the formula δω = ∇ · ω∗,
for any 1-forms ω.
The Markov (or heat) semigroup. Let Ψ : M 7→ R be a fixed C∞-function. Let µ(dx) = e−Ψdx and since M is compact we can assume that µ is a probability measure. Let Lf =
∆f − ∇Ψ · ∇f , for any smooth functions f . Since the manifold is compact, the operator L defines a unique Markov semigroup (Pt)t>0 on L2(µ) and it is called a Markov generator. This
semigroup is symmetric in L2(µ) and for any f , Ptf is a solution of the equation ∂tu = Lu with
f as the initial condition. If f is a probability density (with respect to µ) then for all t > 0, Ptf µ remains a probability measure. Finally, for any smooth functions f and g on M ,
Z M f Lgdµ = Z MgLf µ = − Z M∇f · ∇gdµ = − Z M Γ(f, g)dµ,
where Γ is Carr´e du champ operator defined on functions, Γ(f, g) = ∇f · ∇g. If X is a vector field, we define ∇Ψ· X = ∇ · X − ∇Ψ · X, and the generator L takes then the form
L = ∇Ψ· ∇.
As before we note by δΨ the divergence operator acting on forms:
δΨω = ∇Ψ· ω∗. (6)
It satisfies the integration by parts formula, for any smooth 1-forms ω and functions f , Z
M
δΨωf dµ = −
Z
where ω · df = ωidfi (the inner product between the two 1-forms).
The triple (M, µ, Γ) is a compact Markov triple as defined in [BGL14, Chap. 3].
The Hodge-de Rham semigroup. Connected to the Markov semigroup (Pt) (associated to
the generator L) one can define the Hodge-de Rham semigroup. As explained in this context in [Bak87] the (modified) Hodge-de Rham operator is acting on smooth 1-forms,
−
→L = (d(0)δ(1) Ψ + δ
(2)
Ψ d(1)), (7)
where d(i) is the differential operator acting on i-forms and δΨ(i) is its adjoint operator in L2(µ) with respect to the usual inner product on i-form : R d(i)ω · η dµ =R ω · δ(i)
Ψ η dµ for any i-forms
ω and (i + 1)-forms η. In the sequel, we omit the exponent (i). For computations, we use the Hodge-de Rham operator toward the Weitzenb¨ock formula, that is for any 1 ≤ i ≤ n,
(−→L ω)i= ∇k∇kωi− ∇∇Ψωi− Ricci(L)(ω∗, ei) = ∇k∇kωi− ∇kΨ∇kωi− Ricci(L)(ω∗, ei), (8)
where Ricci(L) = Riccig+Hess(Ψ) is the so-called Bakry-´Emery tensor (see for instance [Bak87,
Prop. 1.5]). Again, we use the analyst’s convention with respect to the sign. If Ψ = 0, then L = ∆ and −→L is the usual Hodge-de Rham operator noted−→∆.
Since M is compact, the operator −→L induces a semigroup (Rt)t>0 on 1-forms. It is also
symmetric in L2(dµ), for any smooth 1-forms ω and η, Z Mω · − →L η dµ =Z Mη · − →L ω dµ.
Then for any smooth 1-forms ω, Rtω is the solution of ∂tu =−→L u where u : [0, ∞) × M 7→ T M∗
(T M∗ is the cotangent bundle) with ω as the initial condition. The details of the construction
of the Hodge-de Rham semigroup can be found in [Bak87] (see also the references therein). The Hodge-de Rham semigroup is related to the Markov generator by the following commu-tation property: for any smooth 1-forms ω and t > 0,
PtδΨω = δΨRtω. (9)
The easiest way to prove this fundamental identity is to use the definition (7) and the identity δ(1)Ψ δ(2)Ψ = 0.
2.2
The Wasserstein distance
Let P(M) be the set of probability measures in M. The Wasserstein distance between two probability measures ν1, ν2 ∈ P(M), is defined by
W2(ν1, ν2) = inf
Z
M ×M
d2(x, y)dπ(x, y)1/2,
where the infimum runs over all probability measures π in M ×M with marginals ν1 and ν2. We
refer to the monumental work [Vil09] for a reference presentation of this distance, its interplay with the optimal transportation problem and many other issues.
The Wasserstein distance has a dynamical formulation: for any probabilities measure ν1, ν2∈
where the infimum is running over all paths of probabilities (µs)s∈[0,1] and ηs ∈ T M∗ satisfying
in the distributional sense
∂sµs+ δ(µsηs) = 0
µ0 = ν, µ1= ν2.
The Euclidean case has been proved by Benamou-Brenier in [BB00] and the Riemannian case by F. Otto and M. Westdickenberg an the other hand by L. De Pascale, M. S. Gelli and L. Granieri in [OW05, DPGG06].
Let ωs = ηsρs where dµdµs = ρs, in [OW05] the authors state that for any f and g, smooth
probability densities (with respect to µ), W22(f µ, gµ) = inf Z 1 0 Z M |ωs|2 ρs dµds, (10)
where the infimum is running over smooth couples (ρs, ωs)s∈[0,1] where for any s ∈ [0, 1], ρs is a
positive probability density (with respect to µ) and ωs is a 1-form, satisfying
∂sρs+ δΨωs= 0
ρ0 = f, ρ1= g.
This dynamical formulation of optimal transportation has been manly used to get contraction result or Evolutional Variational Inequality (see e.g. [DNS09, DNS12, BGG13]).
2.3
Main result
We can now state the main result of this paper.
Theorem 2.1 (Dimensional contraction in Wasserstein distance) Let (M, g) be a C∞,
n-dimensional, connected and compact Riemannian manifold and let Ψ : M 7→ R be a C∞ -function. We assume that there exits R ∈ R and m > n such that uniformly in M (if m = n then we impose that Ψ = 0),
Ricci(L) > R + 1
m − n∇Ψ ⊗ ∇Ψ. (11)
Then for any smooth probability densities f, g with respect to µ and any t > 0, W22(Ptf µ, Ptgµ) ≤ e−2RtW22(f µ, gµ) −
2 m
Z t
0
e−2R(t−u)Entµ(Pug) − Entµ(Puf )2du, (12)
where Entµ(h) =RMh log hdµ for every probability density h (with respect to µ).
Remark 2.2 When m → ∞ we recover the von Renesse-Sturm result, the exponential contrac-tion of the heat equacontrac-tion with respect to the Wasserstein distance. When Ψ = 0, one can choose m = n, and then the Laplace-Beltrami operator satisfies the condition (11) under a lower bound on the Ricci curvature.
As it will be explained in Remark 3.7, condition (11) is equivalent to the so-called Bakry-´
3
Coercive inequality for the Hodge-de Rham
semi-group
The next proposition state a refined Bochner-Lichnerowicz-Weitzenb¨ock formula for 1-forms. Proposition 3.1 (Refined Bochner-Lichnerowicz-Weitzenb¨ock formula) For any smooth 1-forms α, η and any b ∈ R,
L|η| 2 2 − η · − → L η + 2b α · d|η|2+ 4b2|α|2|η|2 = |∇η + 2b α ⊗ η|2+ Ricci(L)(η∗, η∗), (13) where |∇η + 2b α ⊗ η| has to be understood as the norm of the 2-tensor ∇iηj+ 2b αiηj.
When Ψ = 0, −→∆ be the usual Hodge-de Rham operator. The Weitzenb¨ock (8) implies −
→L ω =−→
∆ω − ∇∇Ψω − Hess(Ψ)(ω∗, ·). (14)
According to Lemma 3.2, equation (13) is equivalent to the identity ∆|η| 2 2 − η · − → ∆η + 2b α · d|η|2+ 4b2|α|2|η|2 = |∇η + 2b α ⊗ η|2+ Riccig(η∗, η∗),
which will be proved in Proposition 3.3. The main difficulty is to prove it for all b ∈ R since when b = 0, equation (13) is classic.
Next Lemma can be found for instance in [Lic58, p. 3]. Lemma 3.2 For any 1-forms η and any 1 ≤ k ≤ n, d|η|22
k= η i∇
kηi.
Proposition 3.3 For any smooth 1-forms α, η and any b ∈ R, ∆|η| 2 2 − η · − → ∆η + 2b α · d|η|2+ 4b2|α|2|η|2 = |∇η + 2b α ⊗ η|2+ Riccig(η∗, η∗). (15) Proof
⊳ The Bochner-Lichnerowicz-Weitzenb¨ock formula (c.f. [Lic58, P. 3]) insures that for any smooth 1-forms ω, ∆|ω| 2 2 − ω · − → ∆ω = |∇ω|2+ Riccig(ω∗, ω∗), (16)
recall that |∇ω|2 = ∇iωjgilgjk∇lωk is the square of the norm of the 2-tensor ∇iωj.
The idea is to change variables into this formula. We would like to prove this result at some x0 ∈ M which is supposed to be fixed. Let
ω = (bg + 1)η + b f α,
where η and α are 1-forms and f, g are actually smooth functions satisfying f (x0) = g(x0) = 0.
First we have
|ω|2 = (bg + 1)2|η|2+ b2f2|α|2+ 2b(bg + 1)f α · η.
Since, for any smooth functions F and G we have ∆(F G) = 2 Γ(F, G) + F ∆G + G∆F , where Γ is the carr´e du champ operator, a straight forward computation gives at the point x0,
∆|ω|2 = 4bΓ(g, |η|2) + 2b|η|2∆g + 2b2|η|2|∇g|2+ ∆|η|2+ 2b2|α|2|∇f |2
Since ω(x0) = η(x0), we have in x0,
2ω ·−→∆ω = 2bη−→∆(gη) + 2η−→∆η + 2bη−→∆(f α). Thanks to Lemma 3.5, at the point x0 (recall that f (x0) = g(x0) = 0),
2ω ·−→∆ω = 2b|η|2∆g + 4bη · ∇∇gη + 2η−→∆η + 2bη · α∆f + 4bη · ∇∇fα.
Moreover, at the point x0,
|∇ω|2= b2|η|2Γ(g) + |∇η|2+ b2Γ(f )|α|2+ 2bη · ∇∇gη + 2bα · ∇∇fη + 2b2η · α Γ(f, g),
and Riccig(ω∗, ω∗) = Riccig(η∗, η∗). At the point x0, the identity (16) applied to ω, becomes
∆|η| 2 2 − η − → ∆η + 2bΓ(g, |η|2) + b2|η|2Γ(g) + b2|α|2Γ(f ) + 2bΓ(f, η · α) + 2b2(η · α)Γ(f, g) − 2bη · ∇∇gη − 2b η · ∇∇fα = b2|η|2Γ(g) + |∇η|2 + b2Γ(f )|α|2+ 2bη · ∇∇gη + 2bα · ∇∇fη + 2b2η · αΓ(g, f ) + Riccig(η∗, η∗)
Let now assume that f and g satisfied moreover df (x0) = η(x0) and dg(x0) = α(x0) (which is
always possible), we obtain
∆|η| 2 2 − η − → ∆η + 2b α · ∇|η|2+ 4b2|η|2|α|2 = 4b2|η|2|α|2− 2b η · ∇(η · α) + 2b η · ∇αη + 2b η · ∇ηα + |∇η|2+ 2b η · ∇αη + 2b α · ∇ηη + Riccig(η∗, η∗).
Then Lemma 3.4 implies that
4b2|η|2|α|2− 2b η · d(η · α) + 2b η · ∇αη + 2b η · ∇ηα + |∇η|2+ 2b η · ∇αη + 2b α · ∇ηη
= 4b2|η|2|α|2+ 4b η · ∇αη + |∇η|2 = |∇η + 2b α ⊗ η|2,
which implies the identity (15). ⊲
The next lemma is classic in the Riemannian context and we skip the proof.
Lemma 3.4 For any 1-forms η and α, d(η · α) = ∇η · α + ∇α · η, e.g. in coordinates (d(η · α))i = αj∇iηj+ ηj∇iαj.
The next result can be seen as a kind of diffusion property as defined in [BGL14]. The result is probably classic but I didn’t find it in the literature.
Lemma 3.5 For any smooth functions f and 1-form ω, −
→∆(f ω) = f−→
∆(ω) + ω∆f + 2∇∇fη. (17)
In other words, for any i, −→∆(f ω)
i = f
− →∆(ω)
Proof
⊳ The Weitzenb¨ock formula (8) insures that − → ∆(f ω) i = ∇ k ∇k(f ω)i− Riccig(f ω∗, ei) = ∇k∇k(f ω)i− f Riccig(ω∗, ei).
Now, using in force the formula ∇k(f ω)i= ωi∂kf + f ∇kωi, we get
∇k∇k(f ω)i = gkl∇l∇k(f ω)i
= gkl(∇k∇lf )ωi+ f gkl∇l∇kωi+ 2gkl∂kf ∇lωi
= ωi∆f + f ∇j∇jωi+ 2∇jf ∇jωi = ωi∆f + f ∇j∇jωi+ 2∇∇fωi,
which implies (17). ⊲
Finally we can state our main estimate.
Corollary 3.6 Assume that there exits R ∈ R and m > n such that uniformly in M (if m = n then we impose that Ψ = 0),
Ricci(L) > R + 1
m − n∇Ψ ⊗ ∇Ψ. (18)
Then for any smooth 1-forms η, α and b ∈ R, (with δΨ defined in (6)),
L|η| 2 2 − η · − → L η + 2bα · d|η|2+ 4b2|α|2|η|2> 1 m(δΨη + 2bα · η) 2 + R|η|2. (19) Proof
⊳ From (13) we only have to verify
|∇η + 2bα ⊗ η|2+ Ricci(L)(η∗, η∗) > 1
m(δΨη + 2bα · η)
2+ R|η|2.
Let x ∈ M and let assume that (ei)1≤i≤n is an orthonormal basis of TxM . Then the left hand
side can be written
|∇η + 2bα ⊗ η|2 = n X i,j=1 (∇iηj+ 2bαiηj)2 > 1 n n X i=1 ∇iηi+ 2bα · η 2 ,
from the Cauchy-Schwartz inequality. So it remains to prove that for any t ∈ R 1 n(t + 2bα · η) 2+ Ricci(L)(η∗, η∗) > 1 m(t − dΨ · η + 2b α · η) 2+ R|η|2, where t = δη =Pn
i=1∇iηi (recall that (ei) is an orthonormal basis). The formula is valid since
m > n and from (18) the discriminant of this two order polynomial function in variable t is non-positive (it doesn’t depend on the parameter b).
As proposed by the referee the last inequality can be stated directly by using the quadratic inequality 1 m − nx 2+ 1 ny 2> 1 m(x + y) 2,
Remark 3.7 (Link with the curvature-dimension condition) The so-called Bakry- ´Emery curvature-dimension condition CD(R, m) for an operator L is satisfied when for every smooth function f , Γ2(f ) > R Γ(f ) + 1 m(Lf ) 2, where Γ2(f ) = Γ2(f, f ) = 1 2 LΓ(f ) − 2Γ(f, Lf ).
The same procedure in the case of closed 1-forms has been stated in [Bak94] (see also [ABC+00, Chap. 5]) in the context of Γ2-calculus. It is proved that if η = df and α = dg, then under the
curvature-dimension inequality CD(R, m), and any b ∈ R, Γ2(f ) + 2bΓ(f, Γ(g)) + 4b2Γ(f )Γ(g) > 1
n(Lf + 2bΓ(f, g))
2+ ρΓ(f ), (20)
which is (19) for closed 1-form. Moreover, inequality (20) for every function f (with b = 0) is equivalent to CD(R, m). Since inequality (19) is a generalization of (20), it is also equivalent to (18) and then to CD(R, m).
We can now give the main estimation of our semigroups:
Theorem 3.8 (Coercive estimation) Assume that there exits R ∈ R, m > n such that (if m = n then we impose that Ψ = 0),
Ricci(L) > R + 1
m − n∇Ψ ⊗ ∇Ψ. (21)
Then for any smooth 1-forms ω and smooth functions g > 0, for any t > 0, |Rtω|2 Ptg ≤ e −2RtP t |ω|2 g −m2 Z t 0 e−2Ru Ptg
PtδΨω − Pu(d(log Pt−ug) · Rt−uω)
2
du. (22)
Proof
⊳ Since M is compact, one can assume that there exists ε > 0 such that g = f + ǫ with f > 0 and then ε → 0 in (22). Let ω be a smooth 1-form and t > 0. For any s ∈ [0, t], we define
Λ(s) = Ps |Rt−sω|2 Pt−sg . For any s ∈ [0, 1], Λ′(s) = Ps L|Rt−sω| 2 Pt−sg − 2Rt−sω · − →L R t−sω Pt−sg + LPt−sg|Rt−sω| 2 (Pt−sg)2 .
Corollary 3.6 appied to α = d log G and b = −1/2, implies Λ′(s) > 2 mPs 1 G δΨη − d(log G) · η 2 + 2RPs |η|2 G . Thanks to the Cauchy-Schwarz inequality,
Ps 1 G δΨη − d(log G) · η 2 > 1 PsG h Ps δΨη − d(log G) · η i2 .
Since PsG = PsPt−sg = Ptg and from (9), PsδΨRt−sω = PtδΨω, the inequality becomes
(Λ(s)e−2Rs)′ > 2 m e−2Rs Ptg h PtδΨω − Ps d(log Pt−sg) · Rt−sω i2 . The integration over s ∈ [0, t] of the previous inequality implies (22). ⊲
4
Proof of Theorem 2.1
⊳ Regularity assumption. Let fε = (Pεf + ε)/(1 + ε) and gε = (Pεg + ε)/(1 + ε), for
ε > 0. The probability measure fεµ (resp. gεµ) converges weakly to f µ (resp. gµ), when
ε → 0 and since M is compact W22(fεµ, gεµ) converges to W2(f µ, gµ). The same is also true
for W2
2(Ptfεµ, Ptgεµ). Thus, on can assume that f and g are two smooth functions satisfying
f, g > ε for some ε > 0.
Contraction inequality. Let f, g > ε, be two smooth functions and let (ρs, ωs)s∈[0,1] be a
smooth couple satisfying
∂sρs+ δΨωs= 0
ρ0= f, ρ1 = g. (23)
For any t > 0, from the commutation property (9), the couple (Ptρs, Rtωs)s∈[0,1] satisfies
∂sPtρs+ δΨRtωs= 0
Ptρ0 = Ptf, Ptρ1 = Ptg. (24)
On can apply Theorem 3.8 to get Z 1 0 Z M |Rtωs|2 Ptρs dµds ≤ e −2RtZ 1 0 Z M |ωs|2 ρs dµds −m2 Z 1 0 Z t 0 e−2Ru Z M
PtδΨωs− Pu(d(log Pt−uρs) · Rt−uωs)2
Ptρs
dµduds. (25) Cauchy-Schwarz inequality implies
Z
M
PtδΨωs− Pu(d(log Pt−uρs) · Rt−uωs)2
Ptρs dµ > 1 R Ptρsdµ hZ M
PtδΨωs− Pu(d(log Pt−uρs) · Rt−uωs)
dµi2=h Z
M
d(log Pt−uρs) · Rt−uωsdµ
sinceR
MPtδΨωsdµ =
R
MδΨωsdµ = 0. Integrating over s ∈ [0, 1], thanks again to the
Cauchy-Schwartz inequality, Z 1 0 hZ M∇(log P t−uρs) · Rt−uωsdµ i2 ds >h Z 1 0 Z M
d(log Pt−uρs) · Rt−uωsdµds
i2
=Entµ(Pt−uf ) − Entµ(Pt−ug)2,
since coming from (24) we have (since M is compact, the next integration by parts is valid) d dsEntµ(Pt−uρs) = d ds Z M
Pt−uρslog Pt−uρsdµ =
Z
M
∂sPt−uρslog Pt−uρsdµ
= − Z
M
δΨRt−uωslog Pt−uρsdµ =
Z
M
Rt−uωs· d(log Pt−uρs)dµ.
The inequality (25) becomes Z 1 0 Z M |Rtωs|2 Ptρs dµds ≤ e −2Rt Z 1 0 Z M |ωs|2 ρs dµds − 2 m Z t 0
e−2RuEntµ(Pt−uf ) − Entµ(Pt−ug)
2 du Thanks to the Brenier-Benamou formulation (10) with respect Ptf µ and Ptgµ and formula (24),
W22(Ptf µ, Ptgµ) ≤ e−2Rt Z 1 0 Z M |ωs|2 ρs dµds − 2 m Z t 0
e−2RuEntµ(Pt−uf ) − Entµ(Pt−ug)
2 du. Taking now the infimum over all couples (ρs, ωs)s∈[0,1]satisfying (23),
W22(Ptf µ, Ptgµ) ≤ e−2RtW22(f µ, gµ) −
2 m
Z t
0
e−2RuEntµ(Pt−uf ) − Entµ(Pt−ug)
2 du, which is the inequality desired changing t − u by u. ⊲
Acknowledgements. The author warmly thank F. Bolley and A. Guillin for stimulating discussions. This research was supported by the French ANR-12-BS01-0019 STAB project. The author would like also to thanks the anonymous referee for carefully reading the manuscript.
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